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### Course: Digital SAT Math>Unit 8

Lesson 5: Nonlinear functions: medium

# Nonlinear functions | Lesson

A guide to nonlinear functions on the digital SAT

## What are nonlinear functions problems?

### What is a function?

What is a function?See video transcript
A function takes an input and produces an output. In function notation, $f\left(x\right)$, $f$ is the name of the function, $x$ is the input variable, and $f\left(x\right)$ is the output.
For example, given $f\left(x\right)=2x+1$, the expression $2x+1$ works as instructions on what to do with the input $x$. In this case, the input $x$ is multiplied by $2$, then $1$ is added to the product.
The input of a function can be a
, an
, or even
. Functions can also be
.
In this lesson, we'll learn to:
1. Evaluate functions algebraically
2. Determine inputs and outputs using tables
3. Evaluate
You can learn anything. Let's do this!

## How do I evaluate functions?

### Evaluate a function given its formula

Worked example: Evaluating functions from equationSee video transcript

### Evaluating functions algebraically and using tables

When we encounter an algebraic function, we can find the value of the function at specific inputs. For example, for $f\left(x\right)=2x+1$, we can calculate $f\left(2\right)$, the output of the function $f$ when its input, $x$, is equal to $2$.

### What are the steps?

To evaluate a function at a specific input value:
1. Plug in the input value for the input variable wherever it appears.
2. Perform the operations specified by the function to calculate the ouput.

Example: If $f\left(x\right)={x}^{3}-9$, what is the value of $f\left(-2\right)$ ?

With algebraic functions, we can evaluate the function using multiple inputs to create multiple input-output pairs. These input-output pairs can be put in a table, as shown below for $f\left(x\right)=2x+1$.
$x$$f\left(x\right)$
$0$$1$
$1$$3$
$2$$5$
$3$$7$
Sometimes, a table of input-output pairs is provided without an algebraic function. Consider the table below.
$x$$g\left(x\right)$
$-2$$7$
$-1$$0$
$0$$4$
$1$$-1$
The table contains four input-output pairs. We can interpret the information in the table as:
• $g\left(-2\right)=7$
• $g\left(-1\right)=0$
• $g\left(0\right)=4$
• $g\left(1\right)=-1$
To evaluate a function using a table:
1. Find the input value you're looking for in the input column (typically the left column with a header of the input variable such as $x$).
2. Find the corresponding output value in the output column.

Example:
$x$$f\left(x\right)$
$0$$3$
$1$$2$
$3$$5$
$4$$3$
Based on the table above, what is the value of $f\left(3\right)$ ?

### Try it!

try: complete a table based on a function
$f\left(x\right)={x}^{2}-3$
Complete the table below.
$x$$f\left(x\right)$
$0$
$1$$-2$
$2$
$4$

## How do I evaluate composite functions?

### Intro to function composition

Intro to composing functionsSee video transcript

### Evaluating composite functions algebraically and using tables

A composite function uses the output of one function as the input of another. For example, for $f\left(g\left(x\right)\right)$:
• $x$ is the input to function $g$.
• $g\left(x\right)$ is the input to function $f$.
As such, composite functions should be worked from the inside out. Order matters when evaluating composite functions: $g\left(f\left(x\right)\right)$ is not the same as $f\left(g\left(x\right)\right)$! For example, for $f\left(x\right)={x}^{2}$ and $g\left(x\right)=x+1$, $f\left(g\left(1\right)\right)=4$, but $g\left(f\left(1\right)\right)=2$.
To evaluate composite functions at a specific input value:
1. Plug in the input value for the input variable wherever it appears in the
.
2. Perform the operations specified by the inner function to calculate the output. This output becomes the input of the
.
3. Plug in the result of Step 2 for the input variable wherever it appears in the outer function.
4. Perform the operations specified by the outer function to calculate the final output.

Example: If $f\left(x\right)=2x+1$ and $g\left(x\right)={x}^{2}-2x+1$, what is the value of $f\left(g\left(-1\right)\right)$ ?

Composite functions can also be evaluated using a table. The table can have an additional column for a total of three: one column for input and two columns for the outputs of two functions. Consider the table for $f\left(x\right)={x}^{2}$ and $g\left(x\right)=x+1$:
$x$$f\left(x\right)$$g\left(x\right)$
$1$$1$$2$
$2$$4$$3$
$3$$9$$4$
$4$$16$$5$
From the table, we can tell that $g\left(1\right)=2$, and $f\left(2\right)=4$. Therefore, $f\left(g\left(1\right)\right)=4$.
To evaluate composite functions at a specific input value given a table:
1. Find the output value for the inner function corresponding to the specific input value. This is also the input value of the outer function.
2. Find the output value for the outer function corresponding to the input of the result of Step 1.

Example:
$x$$f\left(x\right)$$g\left(x\right)$
$1$$3$$0$
$2$$5$$1$
$3$$7$$4$
$4$$9$$9$
$5$$11$$16$
The table above provides the values of functions $f$ and $g$ at several values of $x$. What is the value of $g\left(f\left(2\right)\right)$ ?

### Try it!

try: evaluate composite functions using a table
$x$$p\left(x\right)$
$0$$1$
$1$$2$
$2$$3$
$3$$5$
The table above shows the value of function $p$ at several values of $x$. Since $p\left(2\right)=$
and $p\left(3\right)=$
, the value of $p\left(p\left(2\right)\right)$ is
.

## How do I compose functions?

### Finding composite functions

Finding composite functionsSee video transcript

### Inputting expressions instead of values into functions

In addition to inputting a specific value, we can also input one function into another function, which creates a composite function defined by a single expression.
For example, for $f\left(x\right)={x}^{2}$ and $g\left(x\right)=x+1$, $f\left(g\left(x\right)\right)$ replaces each instance of $x$ in $f$ with $g\left(x\right)$, which is equal to $x+1$: $f\left(g\left(x\right)\right)=\left(x+1{\right)}^{2}$. Inputting an expression into a function, e.g., $f\left(x+1\right),$ works similarly.
A function can also be defined in terms of another function. For example, for $f\left(x\right)={x}^{2}$ and $g\left(x\right)=f\left(x\right)+1$, we can replace the $f\left(x\right)$ in function $g$ with ${x}^{2}$: $g\left(x\right)={x}^{2}+1$.
If you find yourself struggling to rewrite complex functions, you might want to brush up on the Operations with polynomials and Operations with rational expressions skills, which have their own lessons.
To compose two functions:
1. Plug in the expression that defines the inner function wherever the input variable appears in the outer function.
2. Perform the operations specified by the outer function. Combine like terms as needed.

#### Let's look at some examples!

If $f\left(x\right)=x-1$ and $g\left(x\right)={x}^{2}+1$, what is $f\left(g\left(x\right)\right)$ ?

If $g\left(x\right)={x}^{2}+1$, what is $g\left(x-1\right)$ ?

### Try it!

try: identify equivalent function composition
$\begin{array}{rl}f\left(x\right)& ={x}^{2}+3x+7\\ \\ g\left(x\right)& ={x}^{3}\end{array}$
The functions $f$ and $g$ are shown above. Which of the following expressions is equivalent to $g\left(f\left(x\right)\right)$ ?

try: Draw conclusions about the features of two related functions
$\begin{array}{rl}f\left(x\right)& =-{x}^{2}+10\\ \\ g\left(x\right)& =f\left(x\right)-7\end{array}$
The functions $f$ and $g$ are shown above. For the same value of $x$, the value of $g\left(x\right)$ is always
the value of $f\left(x\right)$.
Therefore, if the maximum value of the function $f$ is $10$, then the maximum value of function $g$ is
.

TRY: evaluate a function at integer values
$p\left(x\right)=\frac{2}{x}$
The function $p$ is defined above. What is the value of $p\left(2\right)+p\left(5\right)$ ?

TRY: evaluate a composite function
If $f\left(x\right)=3x+5$ and $g\left(x\right)={x}^{2}-4$, what is the value of $g\left(f\left(1\right)\right)$ ?

TRY: evaluate a composite function using table of values
$x$$h\left(x\right)$
$-3$$7$
$-2$$4$
$1$$0$
$3$$-3$
$4$$3$
The table above shows the value of the function $h$ at certain values of $x$. What is the value of $h\left(h\left(3\right)\right)$ ?

TRY: compose two functions
If $f\left(x\right)={x}^{2}-2x-5$, which of the following is equivalent to $f\left(x+3\right)$ ?

## Want to join the conversation?

• MASTERED Alhamdullilah
• yes AlhamduleAllah
• MASTERED Alhamdullilah. Nice videos❤❤❤❤👍👍👍👍👍
• none of that explain the questions involving graphs in the topic test
• fr man, I can't figure out
• There's no lessons on the function graphs
• there is in the next unit
• This is just Higher-Order Function in programming...
After studied college courses and years of experience in Programming, I still have to take SAT as College Entry Exam, what a none-sense...
• In India, you have to learn Integrals. As we usually take SAT in the last year, I don't think anybody can do badly on SAT.
I also did college courses from MIT at https://ocw.mit.edu/.
• I'm just curious, guys. Are here mostly international students, or Americans as well?